Rainfall Gate and the Skewed Claim
A regional water authority insures smallholder farms against drought. Payouts trigger when seasonal rainfall at a monitoring station falls below 300 mm. Historical records for the station span 40 years; the mean is 342 mm and the standard deviation is 58 mm. The authority's actuary assumes rainfall is normally distributed and prices the contract so that the trigger probability is about 0.23. A consultant notes that the 40-year sample has a skewness of -1.4 and that three of the driest five years occurred in the last decade. The board wants to know whether the normal assumption is defensible, and if not, how the premium should be adjusted before the next enrolment window opens in six weeks.
1. Assess whether the normal model is appropriate for pricing this trigger, and quantify the direction of the pricing error if it is not.
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The normal model is questionable on two counts. First, the sample skewness of -1.4 is large relative to its standard error, roughly sqrt(6/40) ≈ 0.39, so the statistic is about 3.6 standard errors from zero; this is strong evidence against symmetry. Second, the clustering of dry years suggests serial dependence, which violates the independence assumed when converting a 40-year record into an effective sample size.
Under the normal model the trigger probability is P(X < 300) = Φ((300 − 342)/58) = Φ(−0.724) ≈ 0.234, matching the actuary's figure. But with negative skew the left tail is heavier than the normal tail at the same mean and variance. A rough check with a skew-normal approximation, or with the empirical distribution, will place P(X < 300) materially above 0.234 - plausibly in the 0.30–0.35 range given the observed dry-year frequency of 3 in 10 recently.
The pricing implication is that the contract is underpriced: the true trigger probability exceeds the assumed 0.23, so expected payout is understated. The board should not simply swap in a normal fit to the recent decade, because that discards information and inflates variance. A defensible route is to fit a skewed distribution or use the empirical distribution with a block bootstrap that preserves serial dependence, then reprice. The six-week window is enough for this; a full Bayesian revision is not needed.
2. The board proposes dropping the 30 oldest years and pricing on the recent decade only. Evaluate this proposal.
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Dropping the oldest 30 years leaves 10 observations. The standard error of a trigger probability estimated from 10 years is roughly sqrt(p(1−p)/10); at p ≈ 0.3 this is about 0.145, so a 95% interval spans roughly 0.02 to 0.58. That is far too wide to price a contract, and it makes the premium hostage to a handful of years.
The proposal also confuses two questions. If the concern is a genuine trend in the rainfall process, the right response is to model the trend explicitly - for example a regression of seasonal rainfall on year, or a time-series model with a drift term - and then derive the trigger probability from the fitted model with its uncertainty. If the concern is only that the recent decade looks dry by chance, truncating the record is pure selection and biases the estimate downward.
A workable compromise: fit a model on all 40 years that allows a change in mean, test whether the change is significant, and if it is, use the post-change mean with a variance estimated from the full record or by pooling. Report the premium with an interval rather than a point. The board should be told that a decade of data cannot support a precise premium, and that the honest output is a range plus a stated assumption about trend.